Question: Let $f(x) = -3x^{2}+8x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}+8x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = 8, c = 9$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot -3 \cdot 9}}{2 \cdot -3}$ $ x = \dfrac{-8 \pm \sqrt{172}}{-6}$ $ x = \dfrac{-8 \pm 2\sqrt{43}}{-6}$ $x =\dfrac{-4 \pm \sqrt{43}}{-3}$